Download Section 3 Chapter 1

Dave Shattuck
© University of Houston
Circuits and Electronics, by Dave Shattuck
Chapter 1
Section 3
Current and Voltage
The two fundamental quantities of interest to us in circuit analysis are current and
voltage. We need to understand these quantities to be able to understand the field of circuit
analysis. Although you may have seen these quantities before, and have heard their definitions
in other courses, we will start with them again, as if you had never seen them before. It is very
important that we get them clearly defined from the beginning.
We will begin with current. Current is defined as the net flow of positive charges,
through some plane in an electrical device, per time. First, let’s discuss the charges. Electrical
charges can be positive or negative. While we know that in most cases the charges that move are
electrons, which have a negative charge, we will generally not be concerned about whether
negative or positive charges are moving. While the charges that move may be important in some
special applications, very often the nature of the charges, positive or negative, does not matter.
We will think in terms of the movement of positive charges. Thus, when we say the net flow of
positive charges, we mean that positive charges flowing from left to right have the same net
result as negative charges flowing from right to left. There are other fields where current is
defined in a different way, often referred to as electron flow. In Electrical Engineering, the
definition is in terms of the flow of positive charges.
Second, let’s talk about flow through a plane in an electrical device. We use the term
device to refer to almost anything that could have charges moving through it. Thus, although we
have not formally defined them yet, we could be talking about things such as resistors or sources.
However, it is important to note that we could talk about the flow through almost anything. We
could be interested the current through the filament of a light bulb, through a pen or a pencil, or
even through your own body. The concept that is important here is that in anything; when we
talk about current through that thing, there must be a plane that could be imagined as cutting
across that object. We then imagine that we could count the charges going past that plane, per
time. That net number of positive charges, going through that plane, is current.
Definition: Current is the net flow of positive charges through some plane
in an electrical device, per time.
Shattuck, Circuits and Electronics
Some
electrical
device
Charge moving
through device
+
+
+
+
+
+
+
+
Plane through this
device
Figure 1.3.1. Current Flowing Through a Device. This is a conceptual picture of current flow
through a device. The device could be anything that has charges moving through it. If we were
to count the net number of charges to flow through a plane that cuts through the device, as shown
here, as a function of time, that would be current. Note that if we had negative charges moving
to the left, this would be the same net result as positive charges moving to the right.
We will use the letter i for the current variable. Historically, this is due the term intensity
that had been used for this quantity, but we should not use this term any more, since it could be
confused with other quantities with that name. We can now show the formal mathematical
definition of current, which is
i
dq
,
dt
(Eq. 1.1)
where q is the charge, and t is time. Here, defining current as a differential equation brings out
the part of the definition where we recognize current as the number of charges per time. This
mathematical definition, while also useful, does not make clear that we are talking about the
charges moving through a plane in some object, nor that we are talking about the net flow of
positive charges. We should think of the definition in words, and the definition in mathematical
terms, as being complementary views of the same thing.
Now, we immediately need to talk about a key issue in all of our discussions of these two
fundamental quantities of current and voltage. This key issue is that of polarity. By this, we
mean, which way does it go? For current the specific question is, which way are the positive
charges flowing? The answer to this question may be extremely important. Indeed, in some
cases the polarity of the quantity may be more important than the magnitude or size of the
quantity. We will keep track of polarity by using what we call reference polarities and actual
polarities. After we have defined voltage, we will describe what reference polarities and actual
polarities are for both quantities, and explain how they are defined and used.
The next issue is that of units. While Eq. 1.1 does not have units shown in it, in
accordance with our plan as laid out in Section 1.2, it is reasonable to introduce the units for
current here. We use the unit [Amperes] for current, where an [Ampere] is defined as a
[Coulomb] per [second]. We abbreviate Amperes as [A], Coulombs as [C], and seconds as [s].
So, we have
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1[A] 
1[C]
.
1[s]
(Eq. 1.2)
Note that the case (uppercase or lowercase) is important in these units. For example, the
uppercase version of the letter s, [S], is used for the units of conductance, which will be
introduced later. A [Coulomb], or [C], is a defined number of electron charges, approximately
6.24 x 1018 electron charges.
Hydraulic Analogy for Current
It is often useful to look at analogies in our attempts to understand certain
concepts. The concept of current is a very important one, so we will introduce a
hydraulic analogy to current. Before we do so, however, it is important to note
that these analogies are always limited; not everything in the analogy is
completely the same for both of the analogous parts. Still, many people find these
analogies useful.
In this case, in a water system, or hydraulic system, the volume flow rate
would be analogous to current. Think of water flow in a pipe. If you were to
imagine having a plane that cuts across this pipe in some location, you can
measure the volume of water that flowed through that plane per time. For
example, in a water pipe you might have 3 cubic meters of water flow past that
plane per second. This volume flow rate, in some ways, would be analogous to
current. It is a rate, in that a given amount moves in a given time. In addition, it
is about the amount that flows past a specific plane in some object.
However, this analogy breaks down when we consider that we have
positive and negative charges, but we really do not have positive and negative
water. Someone might suggest that we have air in a water pipe, which might be
thought of as negative water, but that air does not have negative mass, it has zero
mass. The point here is that in all analogies, we should use them to help us
understand, but we should not expect the analogy to be perfect in all respects.
Figure 1.3.2. Water Flow Analogy to Current. The volume flow rate of water through a pipe can
be thought of as being analogous to current. If you were to have a plane cut across a pipe, then
the volume flow rate past that point could be measured. That is analogous to the number of
charges going past a plane in an electrical device.
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Next, we will discuss voltage. Voltage is defined as the change in potential energy
between two points, caused by electromagnetic fields, per charge. First, it is extremely important
to note that voltage is always about two points. It is meaningless to speak of the voltage at a
point, unless a second point is understood. You may hear someone speak about the voltage at a
point, but if that is meaningful, it is only because the second point is assumed to be known, and
to be clear to everyone who is listening. Much later in this book, we will introduce situations
where this second point is assumed to be known, in an agreed upon way. Until then, we will be
very careful to always specifically indicate the two points in question.
Second, let us consider what we mean by potential energy. Potential energy is a concept
that we have seen in other fields. This can be a difficult concept, and one that may take some
time to get comfortable with. The idea is that an object has the opportunity, or potential, to do
work in the future. We may think of the opportunity for charges to do work, by considering the
diagrams in Figures 1.3.3 and 1.3.4. In these diagrams, we illustrate the idea that charges in an
electric field have the ability to do work in the future, and therefore have potential energy. This
potential energy changes with the size of the charge. The change in this potential energy
between two points, per charge, is called voltage.
Positive test charges
placed in the electric
field produced by other
charges around it
Large collection of
positive charges
+
Large collection of
negative charges
-
+
Force on these
positive test
charges
Figure 1.3.3. Forces on Test Charges in Electric Field. In this diagram, we have a large
collection of positive charges in one place, and a large collection of negative charges somewhere
else nearby. Some positive test charges that might be placed between these two collections
would have a force on them due to the electric fields from these two collections. Thus, these test
charges have the potential to do work, and have potential energy. The amount of potential
energy is proportional to the size of the test charge.
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Large collection of
positive charges
+
Positive test charges,
position 1
+
+
motion
Large collection of
negative charges
-
Positive test charges,
position 2
Figure 1.3.4. Change in Potential Energy Between Two Points. In this diagram, we look at the
movement of some positive test charges between two points, called position 1 and position 2.
Between these two points, we have a change in potential energy. We call this change in potential
energy voltage. We say that if we look at position 1, with respect to position 2, position 1 had
more potential energy. We call this a positive voltage at position 1 with respect to position 2. In
this case, we could also say that we have a negative voltage at position 2 with respect to position
1.
Definition: Voltage is the change in potential energy between two points,
per charge.
We will use the letter v for the voltage variable. We can now show the formal
mathematical definition of voltage, which is
v
dw
,
dq
(Eq. 1.3)
where q is the charge, and w is energy. Here, defining voltage as a differential equation brings
out the part of the definition where we recognize voltage as the change in energy with respect to
charge. This mathematical definition, while useful, does not make clear that we are talking about
the two points, nor that we are talking about positive charges. As before, these two definitions
are complementary views of the same thing.
The polarity of the voltage is just as important as it was with current. Again, we will
discuss polarities in detail, in terms of reference polarities and actual polarities, later in this
section.
The next issue is that of units. We use the unit [Volts] for voltage, where a [Volt] is
defined as a [Joule] per [Coulomb]. We abbreviated Volts as [V], Coulombs as [C], and Joules
as [J]. So, we have
1[V] 
1[J]
.
1[C]
(Eq. 1.4)
As before, the case (uppercase or lowercase) is important in these units.
One way we can make the definition clearer is to state it in another way. So, we could
say the following:
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Shattuck, Circuits and Electronics
One [Joule] of energy is lost from an electric system when one [Coulomb]
of positive charges moves from one potential to another potential that is
one [Volt] lower.This statement indicates that when we are talking about
voltage, we are typically talking about energy changes for a positive charge. It is important to
note that if we were talking about negative charges, the energy changes would be opposite. This
should make sense. If we were moving a positive charge in a direction where it is opposed by
the electromagnetic field, a negative charge moving in the same direction would be assisted in its
motion by that same field.
Hydraulic Analogy for Voltage
There is a similar analogy for voltage in a hydraulic system. In this case,
in a water system, the change in height would be analogous to voltage. Water has
mass, and there is an assumption here, that we operating in the presence of a
gravitational field. Of course, this is generally true when we are not in space.
Therefore, when you take water and move it from one place to higher place, the
water will have gained potential energy. Thus, there was a change in potential
energy between those two places. This change in potential energy between two
places is analogous to voltage.
Again, this analogy breaks down when we consider that we have positive
and negative charges, but we really do not have positive and negative water. We
don’t have anything that has negative mass. The point here is that in all
analogies, we should use them to help us understand, but we should not expect the
analogy to be perfect in all respects.
Change in height
means change in
potential energy
of the water
Figure 1.3.5. Analogy of Voltage to Change in Height. The change in potential energy between
water at one height and another height in a hydraulic system is a result of the gravitational field,
and is a function of how much water you have. This change in potential energy between two
points is analogous to voltage.
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Shattuck, Circuits and Electronics
Reference Polarities and Actual Polarities
We wish to be able to talk about, solve for, and indicate clearly the directions for both
current and voltage. For current, we wish to be clear about which direction positive charges are
moving. For voltage, we wish to be clear about which direction would result in a positive charge
losing energy. These directions we have just described are what we will call actual polarities.
The actual polarity is the direction which things are. This is relatively straightforward.
However, to get to the actual polarity, we use an approach involving something called reference
polarities. A reference polarity is a direction chosen for the purposes of keeping track. A
reference polarity is an arbitrary choice. One can pick a reference polarity in any way one
wishes. Then, with any quantity with a reference polarity, the sign of the value of that quantity
tells you the actual polarity.
This concept of reference polarities can be difficult at the beginning, until you become
comfortable with this approach. Again, to illustrate this idea we will turn again to an analogy to
this approach. The analogy we choose here is to the directions on a compass. More or less
arbitrarily, in aircraft navigation, north has been chosen as a reference direction. Having made
this choice, one can then say to a pilot of an airplane to go in the direction of 135°. This is now
unambiguous, and will be interpreted as indicating that the pilot should fly southeast. This is
clear because southeast is 135°, clockwise, from north. The point to be made here is that this
direction is only clear because a reference direction, north, has been chosen.
The actual polarity is the direction something is actually going. The reference polarity is
a direction chosen for the purposes of keeping track. We have only two possible directions for
current and voltage. This is in contrast to our example with the compass, where there are
essentially an infinite number of possible directions to go. Since there are only two possible
directions for each quantity, this means that we can indicate the polarity using positive and
negative signs. Thus, if we have a reference polarity defined, and we know the sign of the value
of that quantity, we can get the actual polarity. As another more direct example, suppose we
pick our reference direction for motion as ‘up’. Then, let us imagine that we say that the distance
we go ‘up’ is –5[feet]. We know then, that we have moved an actual distance of +5[feet] down.
We will do a similar thing for voltage and current. We have an advantage in these cases,
in that voltage, being the change in potential energy between two points, has only two possible
polarities. Similarly, for current, charge can move through a plane in only two different
directions. Thus, the approach we use is as follows.
We pick a reference polarity for a quantity, which is a voltage or a current.
Then, the sign of the number that goes with that quantity indicates whether the
actual polarity is in that same direction as that reference polarity, or the opposite
direction from that reference polarity. If the sign of the quantity is positive, the
actual polarity is the same as the reference polarity. If the sign of the quantity is
negative, the actual polarity is in the opposite direction from the reference
polarity.
The key to this process is that the reference polarity must be chosen, and the choice must
be clear to all. We say that the choice of a reference polarity, along with the choice of a name,
defines a voltage or a current. As with all variables, these variables must be defined, or else they
are of very limited use. If the defining has not yet been done when we encounter a variable, we
must define it before we use it. Generally, we can define such variables in any polarity we wish.
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Shattuck, Circuits and Electronics
It is not possible to make a mistake picking a reference polarity. The only mistake we can make
in defining reference polarities is to fail to do it. We show how we will define voltages and
currents in the figures that follow.
To define a current, and to indicate the reference polarity for this current, we place an
arrow next to the device or object that has the current going through it. Next to that arrow, we
place a current variable, which should be the letter i, typically with a subscript. Alternatively,
the arrow can have a number with units next to it. It is also acceptable to have both the current
variable and the number with units as the label for the arrow. See Figure 1.3.6 for examples for
these options.
iA
-30[mA]
Object
Object
a) variable
b) value
with units
iA = -30[mA]
Object
c) variable
and value
with units
Figure 1.3.6. Options for Defining Currents. In each of these options, we are given the reference
polarity from the arrow. The sign of the number associated with that polarity, tells us what the
actual polarity is. In cases b) and c), since we know the sign of the value, we know that the
actual polarity of the current is from right to left. In other words, positive charges are flowing
from right to left.
To define a voltage, and to indicate the reference polarity for this voltage, we place a +
sign and a - sign at the two points across which the voltage is defined. On an imaginary line
between these two signs, we place a voltage variable, which should be the letter v, typically with
a subscript. Alternatively, we can place a number with units next to it, in this location. It is also
acceptable to have both the current variable and the number with units. These should be along a
line between the signs, but the line itself should not be drawn. See Figure 1.3.7 for examples for
these options.
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Shattuck, Circuits and Electronics
+
vA
-
+ -7[V]
-
Object
Object
a) variable
b) value
with units
vA =
+ -7[V]
Object
c) variable
and value
with units
Figure 1.3.7. Options for Defining Voltages. In each of these options, we are given the
reference polarity from the + sign and the - sign that are shown. The sign of the number
associated with that polarity, tells us what the actual polarity is. In cases b) and c), since we
know the sign of the value, we know that the actual polarity of the voltage is at the right with
respect to the left. In other words, the right hand side has a higher potential energy.
We should note that the variables shown are given as lowercase variables, with uppercase
subscripts. This is appropriate, and you should get into the habit of doing this. The reason for
this is difficult to explain at this point, but we will lay out the reasons in a later chapter.
Before leaving the subject of current and voltage, it is good to note the issues of path
dependence for these two variables. Current is a path dependent variable. The value of the
current for any object depends on the path being one that goes through that object. This is why
we use the preposition “through” when we talk about currents, as in “this is the value of the
current through the wire”. In contrast, voltage is a path independent variable. The value of the
voltage between at any two points does not depend on the path that one has taken to get between
those two points. This is why we use the preposition “across” when we talk about voltages, as in
“this is the value of the voltage across the gap”. See Figure 1.3.8 for an example, in terms of our
hydraulic analogies.
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Shattuck, Circuits and Electronics
Two Pipes Analogy
This diagram is intended to
show a water pipe that
breaks into two parts and
then combines again. The
size of the blue arrows are
intended to reflect the
amount of water flow at
that point.
Like flow rate,
current is path
dependent.
Like height, voltage
is path independent.
The height
between two
points does
not change
as you go
through the
two pipes.
Flow rate in the
smaller pipe
is less than it is
in the
larger pipe.
Height
Figure 1.3.8. Hydraulic Analogies to Illustrate Path Dependence. If we look at the flow rate
through two pipes, we note that the flow rate is different for the two pipes. This is because flow
rate, as with current, is a path dependent variable. If we look at the change in height across the
two pipes, which are connected together at each end, we note that the change in height is the
same for each pipe. This is because the change in height between two points is path
independent, in the same way as voltage is path independent.
It will be very important that we understand voltage and current, and the appropriate way
to define them and to talk about them. It will be crucial for the remainder of the text, and indeed
for your future with the field itself.
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